The Relationship between Postbuckling Behavior at Coincident Branching Points and the Geometry of an Umbilic Point of the Energy Surface

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Abstract
It is known that the initial postbuckling behavior of elastic structures is determined by the energy function of the critical equilibrium state itself. It has further been shown that if this energy function is expanded as a Taylor series about the critical point, it is the cubic term which affects the classification of postbuckling behavior when it does not vanish. Here, simultaneous buckling in two modes, arising from the coincidence of two branching points, is considered. We show that the potential function at a small distance from the critical branching point can be analyzed using the algebraic theory of surfaces in R3. In suitably normalized coordinates the lines of curvature of this energy

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